Consider a cylindrical tank completely filled with water of height $1.6 \ m$ and cross-sectional area $0.5 \ m^2$. There is a hole in its side at a height of $90 \ cm$ from the bottom. Assume the cross-sectional area of the hole to be negligibly small compared to the cross-sectional area of the water tank. If a load of $50 \ kg$ is applied on the upper surface of water in the tank, then at the moment the hole is opened, the velocity of water coming out is: ($g = 10 \ m/s^2$)
4 m/s
Calculating Efflux Velocity Under Load
This problem involves calculating the speed at which water exits a hole in a cylindrical tank. The calculation must account for both the height of the water above the hole and the additional pressure applied to the water's surface.
Given Parameters
- Tank Height, $H = 1.6 \ m$
- Cross-sectional Area of Tank, $A = 0.5 \ m^2$
- Height of Hole from Bottom, $h_{hole} = 90 \ cm = 0.9 \ m$
- Mass on Upper Surface, $M = 50 \ kg$
- Acceleration due to Gravity, $g = 10 \ m/s^2$
- Density of Water, $\rho \approx 1000 \ kg/m^3$
Determining Applied Pressure and Effective Height
An applied load creates additional pressure on the water surface, increasing the effective height driving the flow.
- Calculate the force exerted by the load:
$ F = M \times g $
$ F = 50 \ kg \times 10 \ m/s^2 = 500 \ N $
- Calculate the pressure applied on the water surface:
$ P_{applied} = \frac{F}{A} $
$ P_{applied} = \frac{500 \ N}{0.5 \ m^2} = 1000 \ Pa $
- Determine the equivalent height of water ($h_{applied}$) corresponding to this pressure:
$ P_{applied} = \rho \times g \times h_{applied} $
$ 1000 \ Pa = 1000 \ kg/m^3 \times 10 \ m/s^2 \times h_{applied} $
$ h_{applied} = \frac{1000}{1000 \times 10} = 0.1 \ m $
- Calculate the actual height of water above the hole:
$ h_{actual} = H - h_{hole} $
$ h_{actual} = 1.6 \ m - 0.9 \ m = 0.7 \ m $
- Find the total effective height driving the flow:
$ h_{eff} = h_{actual} + h_{applied} $
$ h_{eff} = 0.7 \ m + 0.1 \ m = 0.8 \ m $
Calculating Efflux Velocity
Use the modified Torricelli's theorem, which relates efflux velocity to the effective height of the fluid.
The formula for efflux velocity ($v$) is:
$ v = \sqrt{2 \times g \times h_{eff}} $
Substitute the values:
$ v = \sqrt{2 \times 10 \ m/s^2 \times 0.8 \ m} $
$ v = \sqrt{16} \ m/s $
$ v = 4 \ m/s $
The velocity of water coming out of the hole is 4 m/s.
Similar Questions
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is
Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2m, then they will collide after __________ seconds.
A 4.0 cm long straight wire carrying a current of 8A is placed perpendicular to a uniform magnetic field of strength 0.15 T. The magnetic force on the wire is __________ mN.
- The equation for real gas is given by $\left(P+\frac{a}{V^2}\right) (V-b)=RT$, where P,V,T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $ab^{-2}$ is equivalent to that of :
- Match List - I with List - II.Choose the correct answer from the options given below :
List - I List - II (A) Coefficient of viscosity (I) $[ML^0T^{-3}]$ (B) Intensity of wave (II) $[ML^{-2}T^{-2}]$ (C) Pressure gradient (III) $[M^{-1}LT^2]$ (D) Compressibility (IV) $[ML^{-1}T^{-1}]$
Important Questions from Mechanics
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
